1. Field of the Invention
The present invention relates to a method for measuring a curved surface of a workpiece, particularly to a method for measuring a curved surface of a workpiece so as to avoid the interference between the workpiece and a measuring probe.
2. Description of Related Art
Workpieces that have curved surfaces are of great variety and generally the machining of such curved surface of a workpiece is difficult to do. Therefore, the measurement of a curved surface of a workpiece with a high degree of accuracy is strongly demanded.
A gear is cited as a representative example of a workpiece having such a curved surface and, particularly in the final speed reducer of an automobile or the like, gears such as spiral bevel gears and hypoid gears having tooth flanks of curved surfaces are frequently used as gears to change the direction of a rotation axis and reduce the speed in the transmission of rotation power.
A spiral bevel gear is configured so that a ring gear and a pinion engage with each other and their axes intersect with each other on the same plane as shown in FIG. 20.
In contrast, in the case of a hypoid gear, although a ring gear and a pinion engage with each other likewise, their axes do not intersect with each other on the same plane and so-called offset is incorporated. In this regard, the features of a hypoid gear are that it has a high degree-of-freedom of spatial allocation in a power transmission system and moreover allows smoother rotation, quieter operation and also higher tooth strength than a spiral bevel gear.
Such gears are required to prevent wear and noise making from the standpoint of power transmission, thus high accuracy machining is inevitably required, and resultantly a measurement method with a high degree of accuracy is longed for.
However, the tooth flank of such a gear curves in both the tooth trace and tooth depth directions and therefore a problem here is that both the machining and measurement of the gear are difficult to do.
For example, in the case of the ring gear of a hypoid gear having the basic parameters shown in FIG. 21, gear generation is executed with a gear generator based on the theoretical machine setting parameters shown in FIG. 22. The same is true in the case of a pinion and, if nothing is done, a gear pair showing good tooth bearing can not always be obtained due to the mechanical errors and the like of a gear generator.
In this light, a gear pair showing good tooth bearing, which is the final target, is secured by repeated trial gear cuttings (iterative gear cutting operation for better tooth bearings through correcting machine setting parameters) depending on the experience and intuition of a field technician while observing the tooth bearing which is a contact imprint between tooth flanks. This procedure is called “development” (iterative operation for having good tooth bearing).
If the development track (which machine setting parameters is corrected by what degree) can be clarified reversely from a gear produced through the aforementioned procedure, then the influence of mechanical errors intrinsic to each gear generator can be avoided. With the aim of that, methods for estimating machine setting parameters have been under study.
An example of gear cutting principles of the ring gear of a hypoid gear is explained referring to FIG. 23.
A cutter 1 of a gear generator is supported by a cradle 2 so as to be rotatable around the cutter axis zc.
Meanwhile, the base material of a ring gear blank as a workpiece W is supported by a workpiece head 3 so as to be rotatable but the workpiece W does not rotate and is fixed while a tooth of the ring gear is in cutting operation.
The coordinate system shown in FIG. 23 includes: a machine coordinate system consisting of an origin point Om being the center of a machine, cradle axis (zm axis), H axis (ym axis) and V axis (xm axis: an axis passing through the origin point Om and being perpendicular to the cradle axis (zm axis) and the H axis); and a cutter coordinate system consisting of an origin point Oc being the center of a cutter and xc, yc and zc axes (refer to FIG. 25).
In addition, with regard to a workpiece W, there is a gear coordinate system consisting of an origin point Og being the center of a gear and xg, yg and zg axes (refer to FIG. 25).
Here, the workpiece axis zg and the cradle axis zm are on the same plane and the machine center Om coincides with the gear center Og.
With regard to the mutual relationship among the coordinate systems on the VH plane, as shown in FIG. 24, the offset between the V axis (xm axis) and the xc axis is represented by Hg and the offset between the H axis (ym axis) and the yc axis is represented by Vg. Here, Xgc shows the position vector on the locus of a cutting blade edge formed when the cutting blade edge of a cutter 1 rotates around the cutter center Oc.
The mutual relationship among the coordinate systems on the ym-zm plane is as shown in FIG. 25. Here, the distance from the reference plane Wb of a workpiece to the gear center Og (V axis) is represented by Lg and a machine root angle (an angle formed by the ym axis and the zg axis) is represented by kgr.
Under such a configuration, after finishing cutting one tooth of a workpiece W with a cutter 1, the rotation of the cutter 1 stops and the workpiece W is retreated, thereafter the workpiece W is rotated by a predetermined angle around the zg axis and the cutting of the next tooth starts while the cutter 1 is rotated again and the workpiece W is returned to the cutting position. All teeth are cut by repeating above procedure, which effectuates that the position vector Xgc representing the rotation locus of the cutter 1 is transcribed on the workpiece W.
The machine setting parameters of the workpiece W (ring gear) cut in such a way as stated above are estimated in the following manner:
1) With regard to the aforementioned gear tooth flank in which one tooth flank is formed by one curved line, a theoretical tooth flank expression X (u, v, C1, C2, . . . , Cn) is created by mechanistically describing the gear cutting process based on each of the theoretical gear cutting parameters (theoretical machine setting parameters: C1, C2, Cn). (Here, X represents a vector, u does the rotating angle of the cutter 1, and v does the distance from the cutter center Oc to the cutting blade edge.)
2) Measured tooth flank data M is obtained by measuring the tooth flank in terms of three-dimensional coordinates (M is a vector).
Here, Mi, the i-th measured data, is expressed by the expression:Mi=X(ui, vi, C1+ΔC1, C2+ΔC2, . . . , Cn+ΔCn)  (1),and the difference between the measured tooth flank data M and a value given by the theoretical tooth flank expression X is determined as residual (M−X). (Here, ΔC1, ΔC2, . . . , and ΔCn mean the unknown correction amounts of the theoretical machine setting parameters.)M−X(u, v, C1, C2, . . . , Cn)=(ΔC1·∂X/∂C1)+(ΔC2·∂X/∂C2)+ . . . +(ΔCn·∂X/∂Cn)  (2).
3) Such gear cutting parameter Cj+ΔCj as the sum of the square of the residual becomes minimum and the standard deviation at that time are computed by the least-square method for the cases of j=1 to n.
4) The gear cutting parameter Ck which makes the standard deviation to be minimum is searched and Ck+ΔCk is regarded as the estimated value of the gear cutting parameter.
5) The estimated values of the gear cutting parameters other than the k-th are computed likewise by using the estimated value of the gear cutting parameter Ck+ΔCk, and the estimated values of all the gear cutting parameters are computed by further repeating this procedure.
6) In measuring the tooth flank in the three-dimensional coordinate, when the coordinate system of the theoretical tooth flank expression Xg before transforming the coordinate data is defined as Og-xg, yg, zg, and the coordinate system of the coordinate measuring machine is defined as Ot-xt, yt, zt, the coordinate system of the coordinate measuring machine is defined as Ot-xt, yt, zt, one of the coordinate axes of the coordinate measuring machine (for example Z coordinate axis zt) is conformed to the gear axis zg and the pitch cone apex (the origin point Og in the coordinate system of the theoretical tooth flank expression X) is conformed to the origin point Ot in the coordinate system of the coordinate measuring machine. (The locus of the cutting blade edge is transcribed to the workpiece W and therefore it is possible to obtain the theoretical tooth flank expression Xg by transforming the coordinate data of the theoretical expression X that expresses the locus of the cutting blade tip). (Xg is a vector).
7) When an unknown angle formed between another coordinate axis of the coordinate measuring machine (for example X coordinate axis xt) and another coordinate axis of the theoretical tooth flank expression Xg (for example X coordinate axis xg) is defined as Ψ, the result obtained by rotating the theoretical tooth flank expression Xg before transforming around the zt axis by the coordinate transformation matrix C(Ψ) related to the rotation is expressed as follows (C and X are vectors):Xt=C(Ψ)Xg  (3).
Based on this relationship, an angle Ψ can be obtained in addition to the estimated machine setting parameters (C1+ΔC1, C2+ΔC2, . . . , Cn+ΔCn) by the aforementioned method and therefore it becomes possible to transform the theoretical tooth flank expression into a measurement coordinate system. Note that, since Ψ is subordinate to all the gear cutting parameters (C1, C2, . . . , Cn), if the unknowns of C1, C2, . . . Cn and Ψ, namely n+1 in total, are not solved by simultaneous equations, they are solved by applying dual simultaneous equations related to the least-square method to each combination of (Ψ and C1), (Ψ and C2), . . . , (Ψ and Cn), namely n combinations in total.
However, in such a method for estimating machine setting parameters, though it is necessary to obtain measured tooth flank data M by measuring the data at many points on the tooth flank by way of coordinate measuring, a part program cannot be generated due to the fact that the workpiece tooth flank expression in the workpiece coordinate system is unknown, therefore manual measurement has to be applied.
This causes blockage of efficiency improvement in estimating machine setting parameters. Moreover, long time manual measurement is required and therefore the problem here is that the environmental conditions of measurement change due to human body temperature, resulting the dimensions of a workpiece W also change, and measurement with a high degree of accuracy cannot be secured. In addition, even in the case of manual measurement, the coordinate values (theoretical values or true values) of a measurement point are unknown and therefore the workpiece is hardly evaluated.
Further, in the case of measuring the curved surface of a workpiece such as the tooth flank of a spiral bevel gear by using a touch signal probe or a scanning probe, since the measuring plane curves, a problem here is that the stylus of the probe or a spherical contact tip at the tip of the probe may interfere with the gear.